Prove that $\lim_{n\to\infty}\|f_n-f\|_p=\lim_{n\to\infty}\|g_n-g\|_p=0$
Solution 1:
Since $f_n - g_n \to f - g$ $\mu$-a.e. and $\|f_n - g_n\|_p \to \|f - g\|_p$ (by the condition $\|f_n - g_n\|_p = 1 = \|f - g\|$), then $f_n - g_n \xrightarrow{\mathscr{L}^p} f - g$. Similarly, $f_n + g_n \xrightarrow{\mathscr{L}^p} f + g$. Therefore, both $f_n \xrightarrow{\mathscr{L}^p} f$ and $g_n\xrightarrow{\mathscr{L}^p} g$.